if $P_{1}$ and $P_{2}$ are distinct places of equal degree of the function field F/K, and $\sigma$ is a K-field automorphism, such that $\sigma(P_{1})=P_{2}$. then, does $\deg (P_{1}\cap K(x))=\deg (P_{2}\cap K(x))$, where K(x) is the rational function field?
in particular, is this true over the hermitian function field?